Finite Group Databasebeta

← #41006583

Subgroups — #41006583

Order 26973.

488 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22582679C311yesmaximal, characteristic, complemented
2#22582648C311yesmaximal, characteristic, complemented
3#22582643C312yesmaximal, complemented
4#22582639C312yesmaximal, complemented
5#22582637C312yesmaximal, complemented
6C3×#15688514C312yesmaximal, complemented
7#22582551C311yesmaximal, characteristic, complemented
8(C3×C37⋊C3)⋊H(3,𝔽3)C311yesmaximal, characteristic, complemented
9C37×#6910C311yesmaximal, characteristic, complemented
10C33×C37⋊C3C3211yescharacteristic
11#15688532—36no—
12#15688531—33no—
13C3×#11758608C3212yes—
14C3×#11758608C3212yes—
153−1+2×C37⋊C3—33no—
16C3×C32⋊(C37⋊C3)—33no—
17C3×C32⋊(C37⋊C3)C3211yescharacteristic
18C3×C32⋊(C37⋊C3)C3212yes—
19C3×C32⋊(C37⋊C3)C3211yescharacteristic
20#15688519—33no—
21#15688516—36no—
22#15688514—36no—
23#15688514—36no—
24#15688512—36no—
25#15688502—33no—
26C3×3−1+2×C37C3212yes—
27C3×3−1+2×C37C3211yescharacteristic
28C3×H(3,𝔽3)×C37C3211yescharacteristic
29C37×#731—33no—
30C33×C37—31no—
31C33×C37C3311yescharacteristic
32C32×C37⋊C3—31no—
33C32×C37⋊C3—31no—
34C32×C37⋊C3—32no—
35C32×C37⋊C3—31no—
36C32×C37⋊C3—32no—
37C32×C37⋊C3—93no—
38C32×C37⋊C3H(3,𝔽3)13yes—
393−1+2×C37—31no—
403−1+2×C37—32no—
413−1+2×C37—32no—
423−1+2×C37—32no—
433−1+2×C37—32no—
44H(3,𝔽3)×C37—31no—
45H(3,𝔽3)×C37—32no—
46C3×C9×C37—32no—
47C3×C9×C37H(3,𝔽3)13yescomplemented
48C32⋊(C37⋊C3)—31no—
49C32⋊(C37⋊C3)—32no—
50C32⋊(C37⋊C3)—32no—
51C32⋊(C37⋊C3)—32no—
52C32⋊(C37⋊C3)—32no—
53C32⋊(C37⋊C3)—31no—
54C32⋊(C37⋊C3)—32no—
55C32⋊(C37⋊C3)—99no—
56#11758608—32no—
57#11758608—32no—
58#11758608—32no—
59#11758608—32no—
60#11758608—32no—
61#11758608H(3,𝔽3)16yescomplemented
62C3×#66524—32no—
63C3×#66524—32no—
64#11758606—39no—
65#11758605—39no—
66C9×C37⋊C3—39no—
67#10624749—371nomaximal, Sylow
68C32×C37—91no—
69C32×C37—93no—
70C32×C37—91no—
71C32×C37C3×H(3,𝔽3)12yes—
72C32×C37C3×H(3,𝔽3)11yescharacteristic
73C32×C37C3×H(3,𝔽3)11yescharacteristic
74C3×C37⋊C3—33no—
75C3×C37⋊C3—33no—
76C3×C37⋊C3—33no—
77C3×C37⋊C3—92no—
78C3×C37⋊C3—92no—
79C3×C37⋊C3—91no—
80C3×C37⋊C3—96no—
81C3×C37⋊C3—93no—
82C3×C37⋊C3—93no—
83C3×C37⋊C3—91no—
84C3×C37⋊C3—92no—
85C3×C37⋊C3—272no—
86C9×C37—33no—
87C9×C37—92no—
88#66524—36no—
89#66524—92no—
903−1+4—371no—
91#6911—371no—
92#6910—372no—
93#6910C37⋊C311yescharacteristic, complemented
94#6909—372no—
95#6909—372no—
96C3×AΣL(1,𝔽27)—372no—
97#6891—371no—
98C32⋊H(3,𝔽3)—371no—
99C3×C37—31no—
100C3×C37—31no—
101C3×C37—32no—
102C3×C37—271no—
103C3×C37C32⋊H(3,𝔽3)11yescharacteristic
104C37⋊C3—93no—
105C37⋊C3—93no—
106C37⋊C3—271no—
107C37⋊C3—272no—
108C34—371no—
109#737—1116no—
110#737—1113no—
111C3×3−1+2—372no—
112C3×3−1+2—372no—
113C3×3−1+2—1113no—
114C3×3−1+2C3×C37⋊C312yes—
115C3×3−1+2C3×C37⋊C311yescharacteristic
116C3×H(3,𝔽3)—371no—
117C3×H(3,𝔽3)—371no—
118C3×H(3,𝔽3)—372no—
119C3×H(3,𝔽3)—1113no—
120C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
121#732—1113no—
122#731—33no—
123#731—1116no—
124AΣL(1,𝔽27)—1116no—
125AΣL(1,𝔽27)—1116no—
126AΣL(1,𝔽27)—1116no—
127C32⋊C9—1113no—
128C37#1062474911yescharacteristic, complemented, Sylow
129C33—31no—
130C33—373no—
131C33—3333no—
132C33—1112no—
133C33—1111no—
134C33—1111no—
135C33—1111no—
136C33—1112no—
137C33C32×C37⋊C311yescharacteristic
1383−1+2—31no—
1393−1+2—32no—
1403−1+2—32no—
1413−1+2—32no—
1423−1+2—32no—
1433−1+2—376no—
1443−1+2—1119no—
1453−1+2—1119no—
1463−1+2—1112no—
1473−1+2—1112no—
1483−1+2—1112no—
1493−1+2—1112no—
1503−1+2—1112no—
151H(3,𝔽3)—32no—
152H(3,𝔽3)—31no—
153H(3,𝔽3)—3339no—
154H(3,𝔽3)—1112no—
155H(3,𝔽3)—1112no—
156H(3,𝔽3)—1112no—
157H(3,𝔽3)—1111no—
158H(3,𝔽3)—1112no—
159H(3,𝔽3)—1112no—
160H(3,𝔽3)—1111no—
161C3×C9—32no—
162C3×C9—1119no—
163C3×C9—1112no—
164C3×C9—1112no—
165C3×C9C32⋊(C37⋊C3)13yescomplemented
166C32—91no—
167C32—93no—
168C32—91no—
169C32—3333no—
170C32—3336no—
171C32—3332no—
172C32—3332no—
173C32—3333no—
174C32—3331no—
175C32—3331no—
176C32—3332no—
177C32—1113no—
178C32—1113no—
179C32—1113no—
180C32—9992no—
181C32H(3,𝔽3)×C37⋊C312yes—
182C32H(3,𝔽3)×C37⋊C311yescharacteristic
183C32C3×C32⋊(C37⋊C3)11yescharacteristic
184C9—33no—
185C9—92no—
186C9—3332no—
187C9—1116no—
188C3—31no—
189C3—32no—
190C3—31no—
191C3—271no—
192C3—3333no—
193C3—3333no—
194C3—9991no—
195C3—9992no—
196C3(C3×C37⋊C3)⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC3(C3×C37⋊C3)⋊H(3,𝔽3)
commutator, cosocleC33×C37C33
FrattiniC33C32×C37⋊C3
FittingC37×#6910C3
socleC3×C37C32⋊H(3,𝔽3)
hypercenter#6910C37⋊C3
hypocenterC37#10624749
LCSG ▷ C33×C37 ▷ C3×C37 ▷ C37
UCSC1 ◁ C3 ◁ C33 ◁ #6910

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22582679C3semidirect
2. #22582648C3semidirect
3. #22582643C3semidirect
4. #22582639C3semidirect
5. #22582637C3semidirect
6. C3×#15688514C3semidirect
7. #22582551C3semidirect
8. (C3×C37⋊C3)⋊H(3,𝔽3)C3semidirect
9. C37×#6910C3semidirect
47. C3×C9×C37H(3,𝔽3)semidirect
61. #11758608H(3,𝔽3)semidirect
93. #6910C37⋊C3semidirect
128. C37#10624749semidirect
165. C3×C9C32⋊(C37⋊C3)semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225826793
#225826483
#225826433
#225826393
#225826373
C3×#156885143
#225825513
(C3×C37⋊C3)⋊H(3,𝔽3)3
C37×#69103
#1062474937